2 Answers
2

There may be a few instances that are relative to the setting in which they are used. The numbering, I'm confident, will always move in the direction in which game play progresses. I haven't seen, and doubt I ever will, anything other than that.

Some examples...

In a casino seat one (1) will almost always be the seat directly to the left of the dealer, if there is a dealer. This is independent of the "Button".

If there isn't a dealer, the numbering will almost certainly move in the direction of the game flow, starting with one (1) for the SB, two (2) for the BB and so on for the remaining players, until the "Button".

There are other conventions that use, as you said you know, the UTG, HJ, CO, BU, SB, BB labels. UTG can carry extra information such as UTG+1, UTG+2, etc.

Position 1 is the first person to act. In a no-blind game this is the person to the left of the dealer. In a game with one or more blinds, this is the first person to have a choice of acting (fold, call, raise). After the first betting round, postion 1 is to the left of the dealer.